Properties

Label 405.4.e.u
Level $405$
Weight $4$
Character orbit 405.e
Analytic conductor $23.896$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.148347072.2
Defining polynomial: \( x^{6} + 29x^{4} + 223x^{2} + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{4} + 5 \beta_{2} q^{5} + ( - 3 \beta_{5} + 10 \beta_{2} + 5 \beta_1 + 10) q^{7} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 8) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{4} + 5 \beta_{2} q^{5} + ( - 3 \beta_{5} + 10 \beta_{2} + 5 \beta_1 + 10) q^{7} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 8) q^{8} + 5 \beta_{3} q^{10} + (5 \beta_{5} + 17 \beta_{2} - 7 \beta_1 + 17) q^{11} + ( - 13 \beta_{4} + 13 \beta_{3} - 20 \beta_{2} - 13 \beta_1) q^{13} + ( - 11 \beta_{4} + 9 \beta_{3} - 56 \beta_{2} - 9 \beta_1) q^{14} + (9 \beta_{5} + 44 \beta_{2} + 5 \beta_1 + 44) q^{16} + ( - 8 \beta_{5} + 8 \beta_{4} - 8 \beta_{3} - 14) q^{17} + (14 \beta_{5} - 14 \beta_{4} - 10 \beta_{3} - 5) q^{19} + ( - 5 \beta_{5} - 10 \beta_{2} - 5 \beta_1 - 10) q^{20} + (17 \beta_{4} + 20 \beta_{3} + 80 \beta_{2} - 20 \beta_1) q^{22} + (19 \beta_{4} - 15 \beta_{3} + 22 \beta_{2} + 15 \beta_1) q^{23} + ( - 25 \beta_{2} - 25) q^{25} + (13 \beta_{5} - 13 \beta_{4} - 59 \beta_{3} - 104) q^{26} + ( - 11 \beta_{5} + 11 \beta_{4} - 47 \beta_{3} + 12) q^{28} + ( - 13 \beta_{5} + 95 \beta_{2} - 65 \beta_1 + 95) q^{29} + ( - \beta_{4} + 5 \beta_{3} + 211 \beta_{2} - 5 \beta_1) q^{31} + (21 \beta_{4} + 43 \beta_{3} - 96 \beta_{2} - 43 \beta_1) q^{32} + ( - 8 \beta_{5} + 64 \beta_{2} + 38 \beta_1 + 64) q^{34} + (15 \beta_{5} - 15 \beta_{4} - 25 \beta_{3} - 50) q^{35} + ( - 2 \beta_{5} + 2 \beta_{4} - 54 \beta_{3} - 156) q^{37} + (38 \beta_{5} + 128 \beta_{2} - 13 \beta_1 + 128) q^{38} + ( - 5 \beta_{4} + 15 \beta_{3} + 40 \beta_{2} - 15 \beta_1) q^{40} + ( - 62 \beta_{4} - 2 \beta_{3} - 59 \beta_{2} + 2 \beta_1) q^{41} + (8 \beta_{5} + 288 \beta_{2} + 28 \beta_1 + 288) q^{43} + ( - 14 \beta_{5} + 14 \beta_{4} + 38 \beta_{3} - 98) q^{44} + ( - 23 \beta_{5} + 23 \beta_{4} + 75 \beta_{3} + 112) q^{46} + ( - 31 \beta_{5} + 56 \beta_{2} - 67 \beta_1 + 56) q^{47} + (7 \beta_{4} - 11 \beta_{3} + 301 \beta_{2} + 11 \beta_1) q^{49} + ( - 25 \beta_{3} + 25 \beta_1) q^{50} + ( - 19 \beta_{5} + 456 \beta_{2} + 33 \beta_1 + 456) q^{52} + ( - 21 \beta_{5} + 21 \beta_{4} - 53 \beta_{3} - 186) q^{53} + ( - 25 \beta_{5} + 25 \beta_{4} + 35 \beta_{3} - 85) q^{55} + ( - 63 \beta_{5} - 15 \beta_1) q^{56} + (39 \beta_{4} + 4 \beta_{3} + 624 \beta_{2} - 4 \beta_1) q^{58} + (90 \beta_{4} - 58 \beta_{3} - 159 \beta_{2} + 58 \beta_1) q^{59} + ( - 58 \beta_{5} - 6 \beta_{2} - 122 \beta_1 - 6) q^{61} + ( - 3 \beta_{5} + 3 \beta_{4} + 204 \beta_{3} - 48) q^{62} + ( - 13 \beta_{5} + 13 \beta_{4} - 57 \beta_{3} - 120) q^{64} + (65 \beta_{5} + 100 \beta_{2} + 65 \beta_1 + 100) q^{65} + ( - 18 \beta_{4} - 154 \beta_{3} + 38 \beta_{2} + 154 \beta_1) q^{67} + (10 \beta_{4} + 22 \beta_{3} - 284 \beta_{2} - 22 \beta_1) q^{68} + (55 \beta_{5} + 280 \beta_{2} + 45 \beta_1 + 280) q^{70} + (69 \beta_{5} - 69 \beta_{4} - 79 \beta_{3} - 177) q^{71} + ( - 12 \beta_{5} + 12 \beta_{4} + 32 \beta_{3} - 226) q^{73} + (50 \beta_{5} + 536 \beta_{2} + 214 \beta_1 + 536) q^{74} + ( - 23 \beta_{4} + 111 \beta_{3} + 246 \beta_{2} - 111 \beta_1) q^{76} + ( - 98 \beta_{4} - 162 \beta_{3} - 662 \beta_{2} + 162 \beta_1) q^{77} + (54 \beta_{5} + 184 \beta_{2} - 82 \beta_1 + 184) q^{79} + ( - 45 \beta_{5} + 45 \beta_{4} - 25 \beta_{3} - 220) q^{80} + (126 \beta_{5} - 126 \beta_{4} - 181 \beta_{3} + 144) q^{82} + ( - 30 \beta_{5} + 648 \beta_{2} + 210 \beta_1 + 648) q^{83} + ( - 40 \beta_{4} + 40 \beta_{3} - 70 \beta_{2} - 40 \beta_1) q^{85} + ( - 12 \beta_{4} + 332 \beta_{3} - 264 \beta_{2} - 332 \beta_1) q^{86} + (70 \beta_{5} + 232 \beta_{2} - 72 \beta_1 + 232) q^{88} + (3 \beta_{5} - 3 \beta_{4} + 39 \beta_{3} + 297) q^{89} + (161 \beta_{5} - 161 \beta_{4} + 581 \beta_{3} - 216) q^{91} + (31 \beta_{5} - 620 \beta_{2} - 21 \beta_1 - 620) q^{92} + (5 \beta_{4} - 73 \beta_{3} + 608 \beta_{2} + 73 \beta_1) q^{94} + (70 \beta_{4} + 50 \beta_{3} - 25 \beta_{2} - 50 \beta_1) q^{95} + ( - 32 \beta_{5} - 112 \beta_{2} - 336 \beta_1 - 112) q^{97} + ( - 3 \beta_{5} + 3 \beta_{4} + 326 \beta_{3} + 96) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 5 q^{4} - 15 q^{5} + 25 q^{7} + 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - 5 q^{4} - 15 q^{5} + 25 q^{7} + 54 q^{8} - 10 q^{10} + 58 q^{11} + 47 q^{13} + 159 q^{14} + 127 q^{16} - 68 q^{17} - 10 q^{19} - 25 q^{20} - 260 q^{22} - 51 q^{23} - 75 q^{25} - 506 q^{26} + 166 q^{28} + 350 q^{29} - 638 q^{31} + 245 q^{32} + 154 q^{34} - 250 q^{35} - 828 q^{37} + 397 q^{38} - 135 q^{40} + 179 q^{41} + 836 q^{43} - 664 q^{44} + 522 q^{46} + 235 q^{47} - 892 q^{49} + 25 q^{50} + 1335 q^{52} - 1010 q^{53} - 580 q^{55} + 15 q^{56} - 1876 q^{58} + 535 q^{59} + 104 q^{61} - 696 q^{62} - 606 q^{64} + 235 q^{65} + 40 q^{67} + 830 q^{68} + 795 q^{70} - 904 q^{71} - 1420 q^{73} + 1394 q^{74} - 849 q^{76} + 2148 q^{77} + 634 q^{79} - 1270 q^{80} + 1226 q^{82} + 1734 q^{83} + 170 q^{85} + 460 q^{86} + 768 q^{88} + 1704 q^{89} - 2458 q^{91} - 1839 q^{92} - 1751 q^{94} + 25 q^{95} - 76 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 29x^{4} + 223x^{2} + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} + 13\nu + 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 20\nu^{3} + 79\nu - 36 ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} + 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 9\nu^{4} - 20\nu^{3} - 144\nu^{2} + 29\nu - 207 ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 9\nu^{4} - 20\nu^{3} + 144\nu^{2} + 29\nu + 207 ) / 72 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{5} - 13\beta_{4} - 6\beta_{3} - 26\beta_{2} + 12\beta _1 - 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{5} - 4\beta_{4} - 32\beta_{3} + 121 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 181\beta_{5} + 181\beta_{4} + 120\beta_{3} + 578\beta_{2} - 240\beta _1 + 289 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
1.13993i
3.41374i
4.00586i
1.13993i
3.41374i
4.00586i
−1.92514 + 3.33444i 0 −3.41235 5.91036i −2.50000 4.33013i 0 13.1641 22.8009i −4.52526 0 19.2514
136.2 0.663404 1.14905i 0 3.11979 + 5.40363i −2.50000 4.33013i 0 12.0521 20.8749i 18.8932 0 −6.63404
136.3 1.76174 3.05142i 0 −2.20744 3.82340i −2.50000 4.33013i 0 −12.7162 + 22.0252i 12.6321 0 −17.6174
271.1 −1.92514 3.33444i 0 −3.41235 + 5.91036i −2.50000 + 4.33013i 0 13.1641 + 22.8009i −4.52526 0 19.2514
271.2 0.663404 + 1.14905i 0 3.11979 5.40363i −2.50000 + 4.33013i 0 12.0521 + 20.8749i 18.8932 0 −6.63404
271.3 1.76174 + 3.05142i 0 −2.20744 + 3.82340i −2.50000 + 4.33013i 0 −12.7162 22.0252i 12.6321 0 −17.6174
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.4.e.u 6
3.b odd 2 1 405.4.e.s 6
9.c even 3 1 405.4.a.g 3
9.c even 3 1 inner 405.4.e.u 6
9.d odd 6 1 405.4.a.i yes 3
9.d odd 6 1 405.4.e.s 6
45.h odd 6 1 2025.4.a.p 3
45.j even 6 1 2025.4.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.g 3 9.c even 3 1
405.4.a.i yes 3 9.d odd 6 1
405.4.e.s 6 3.b odd 2 1
405.4.e.s 6 9.d odd 6 1
405.4.e.u 6 1.a even 1 1 trivial
405.4.e.u 6 9.c even 3 1 inner
2025.4.a.p 3 45.h odd 6 1
2025.4.a.r 3 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{6} - T_{2}^{5} + 15T_{2}^{4} - 22T_{2}^{3} + 214T_{2}^{2} - 252T_{2} + 324 \) Copy content Toggle raw display
\( T_{7}^{6} - 25T_{7}^{5} + 1273T_{7}^{4} - 16080T_{7}^{3} + 823404T_{7}^{2} - 10458720T_{7} + 260499600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + 15 T^{4} - 22 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} - 25 T^{5} + \cdots + 260499600 \) Copy content Toggle raw display
$11$ \( T^{6} - 58 T^{5} + 4275 T^{4} + \cdots + 9000000 \) Copy content Toggle raw display
$13$ \( T^{6} - 47 T^{5} + \cdots + 137160603904 \) Copy content Toggle raw display
$17$ \( (T^{3} + 34 T^{2} - 2708 T - 90984)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 5 T^{2} - 10777 T - 299645)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 51 T^{5} + \cdots + 1084005816336 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 126287249817600 \) Copy content Toggle raw display
$31$ \( T^{6} + 638 T^{5} + \cdots + 90993741996096 \) Copy content Toggle raw display
$37$ \( (T^{3} + 414 T^{2} + 16032 T - 577760)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 316826721339129 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 349427949912064 \) Copy content Toggle raw display
$47$ \( T^{6} - 235 T^{5} + \cdots + 81096796901376 \) Copy content Toggle raw display
$53$ \( (T^{3} + 505 T^{2} + 35128 T - 1500684)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 498077813969649 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 554264938580224 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 246999193899264 \) Copy content Toggle raw display
$71$ \( (T^{3} + 452 T^{2} - 264923 T - 116183454)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 710 T^{2} + 144236 T + 8707528)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 634 T^{5} + \cdots + 25533374939136 \) Copy content Toggle raw display
$83$ \( T^{6} - 1734 T^{5} + \cdots + 49\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{3} - 852 T^{2} + 220725 T - 17926434)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 1575168 T^{4} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
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